By Lemma REF , \(m^-(a_1,0)\)  can be achieved by \(u^*\in S(a_1)\) . We choose a proper test function \(v\in S(a_2)\)  such that \((u^*,s\star v)\in S(a_1)\times S(a_2)\)  with \(s\in \mathbb {R}\) .
In [1]}, [2]}, it has been proved that
\(m^-(a_1,0)=\inf _{u\in \mathcal {T}^{-}_{a_1,p,\mu _1,\alpha _1}} \mathcal {A}_{p,\mu _1,\alpha _1}(u).\) 
